In this variant, for every point \(v_{i}\)  there are at least \(\delta (m-1)\)  points \(u_{1}, \dots , u_{k}\)  in the configuration such that \(v_{i}\)  and \(u_{j}\)  span a third point, for \(1 \le j \le k\) .
The usual Sylvester-Gallai theorem is the case when \(\delta = 1\) .
Such configurations were first studied by Szemerédi and Trotter [1]}, who proved that if \(\delta \)  is bigger than an absolute constant close to 1, then the configuration has constant dimension.
